getwd()
## [1] "/Users/rene/Desktop/CSULB MS/Fall 2020/STAT 510/Project"
# Setup
library(GGally)
## Loading required package: ggplot2
## Registered S3 method overwritten by 'GGally':
## method from
## +.gg ggplot2
library(leaps)
library(MASS)
dat = read.csv('2019b.csv', header = TRUE)
head(dat)
## Score GDP.per.capita Social.support Healthy.life.expectancy
## 1 7.769 1.340 1.587 0.986
## 2 7.600 1.383 1.573 0.996
## 3 7.554 1.488 1.582 1.028
## 4 7.494 1.380 1.624 1.026
## 5 7.488 1.396 1.522 0.999
## 6 7.480 1.452 1.526 1.052
## Freedom.to.make.life.choices Generosity Perceptions.of.corruption
## 1 0.596 0.153 0.393
## 2 0.592 0.252 0.410
## 3 0.603 0.271 0.341
## 4 0.591 0.354 0.118
## 5 0.557 0.322 0.298
## 6 0.572 0.263 0.343
y = dat$Score
x1 = dat$GDP.per.capita
x2 = dat$Social.support
x3 = dat$Healthy.life.expectancy
x4 = dat$Freedom.to.make.life.choices
x5 = dat$Generosity
x6 = dat$Perceptions.of.corruption
n = nrow(dat)
# Scatter-plot of the data
ggpairs(dat, cardinality_threshold = NULL)
ggpairs(dat, cardinality_threshold = 156)
# Step-wise Regression
# a)F-Test:
mod0 = lm(y~1)
add1(mod0, ~.+x1+x2+x3+x4+x5+x6, test='F')
## Single term additions
##
## Model:
## y ~ 1
## Df Sum of Sq RSS AIC F value Pr(>F)
## <none> 192.051 34.433
## x1 1 121.040 71.011 -118.776 262.4976 < 2.2e-16 ***
## x2 1 115.964 76.087 -108.005 234.7110 < 2.2e-16 ***
## x3 1 116.809 75.242 -109.747 239.0755 < 2.2e-16 ***
## x4 1 61.686 130.365 -24.005 72.8697 1.238e-14 ***
## x5 1 1.104 190.946 35.533 0.8905 0.3468
## x6 1 28.557 163.493 11.319 26.8993 6.654e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
mod1 = update(mod0, ~.+x1)
add1(mod1,~.+x2+x3+x4+x5+x6, test='F')
## Single term additions
##
## Model:
## y ~ x1
## Df Sum of Sq RSS AIC F value Pr(>F)
## <none> 71.011 -118.78
## x2 1 14.1076 56.903 -151.33 37.9322 6.177e-09 ***
## x3 1 8.6493 62.361 -137.04 21.2205 8.562e-06 ***
## x4 1 15.8450 55.166 -156.16 43.9454 5.455e-10 ***
## x5 1 3.7379 67.273 -125.21 8.5011 0.004083 **
## x6 1 4.6385 66.372 -127.31 10.6927 0.001329 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
mod2 = update(mod1,~.+x4)
summary(mod2)
##
## Call:
## lm(formula = y ~ x1 + x4)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.3263 -0.4189 0.1303 0.3823 1.1176
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.7503 0.1546 17.789 < 2e-16 ***
## x1 1.8894 0.1308 14.442 < 2e-16 ***
## x4 2.4113 0.3637 6.629 5.45e-10 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.6005 on 153 degrees of freedom
## Multiple R-squared: 0.7128, Adjusted R-squared: 0.709
## F-statistic: 189.8 on 2 and 153 DF, p-value: < 2.2e-16
add1(mod2,~.+x2+x3+x5+x6, test='F')
## Single term additions
##
## Model:
## y ~ x1 + x4
## Df Sum of Sq RSS AIC F value Pr(>F)
## <none> 55.166 -156.16
## x2 1 7.8369 47.329 -178.07 25.1687 1.449e-06 ***
## x3 1 5.7092 49.457 -171.21 17.5465 4.735e-05 ***
## x5 1 0.4566 54.709 -155.46 1.2686 0.2618
## x6 1 0.5451 54.621 -155.71 1.5170 0.2200
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
mod3 = update(mod2,~.+x2)
summary(mod3)
##
## Call:
## lm(formula = y ~ x1 + x4 + x2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.1384 -0.4126 0.0811 0.3958 1.1332
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.0492 0.2004 10.223 < 2e-16 ***
## x1 1.2792 0.1720 7.438 6.98e-12 ***
## x4 1.9442 0.3506 5.545 1.27e-07 ***
## x2 1.1886 0.2369 5.017 1.45e-06 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.558 on 152 degrees of freedom
## Multiple R-squared: 0.7536, Adjusted R-squared: 0.7487
## F-statistic: 154.9 on 3 and 152 DF, p-value: < 2.2e-16
add1(mod3,~.+x3+x5+x6, test='F')
## Single term additions
##
## Model:
## y ~ x1 + x4 + x2
## Df Sum of Sq RSS AIC F value Pr(>F)
## <none> 47.329 -178.07
## x3 1 3.3363 43.993 -187.47 11.4514 0.0009103 ***
## x5 1 0.7866 46.542 -178.68 2.5522 0.1122326
## x6 1 1.6519 45.677 -181.61 5.4608 0.0207613 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
mod4 = update(mod3,~.+x3)
summary(mod4)
##
## Call:
## lm(formula = y ~ x1 + x4 + x2 + x3)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.86584 -0.34594 0.03403 0.43676 1.13076
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.8921 0.1994 9.491 < 2e-16 ***
## x1 0.8105 0.2165 3.745 0.000256 ***
## x4 1.8458 0.3404 5.423 2.28e-07 ***
## x2 1.0166 0.2347 4.331 2.70e-05 ***
## x3 1.1414 0.3373 3.384 0.000910 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.5398 on 151 degrees of freedom
## Multiple R-squared: 0.7709, Adjusted R-squared: 0.7649
## F-statistic: 127 on 4 and 151 DF, p-value: < 2.2e-16
add1(mod4,~.+x5+x6, test='F')
## Single term additions
##
## Model:
## y ~ x1 + x4 + x2 + x3
## Df Sum of Sq RSS AIC F value Pr(>F)
## <none> 43.993 -187.47
## x5 1 0.6661 43.326 -187.85 2.3061 0.13097
## x6 1 1.3053 42.687 -190.17 4.5866 0.03384 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
mod5 = update(mod4,~.+x6)
summary(mod5)
##
## Call:
## lm(formula = y ~ x1 + x4 + x2 + x3 + x6)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.82997 -0.35344 0.05803 0.35977 1.17522
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.8689 0.1973 9.471 < 2e-16 ***
## x1 0.7455 0.2161 3.450 0.000728 ***
## x4 1.5340 0.3666 4.185 4.84e-05 ***
## x2 1.1180 0.2368 4.722 5.33e-06 ***
## x3 1.0840 0.3344 3.241 0.001467 **
## x6 1.1176 0.5218 2.142 0.033839 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.5335 on 150 degrees of freedom
## Multiple R-squared: 0.7777, Adjusted R-squared: 0.7703
## F-statistic: 105 on 5 and 150 DF, p-value: < 2.2e-16
add1(mod5,~.+x5, test='F')
## Single term additions
##
## Model:
## y ~ x1 + x4 + x2 + x3 + x6
## Df Sum of Sq RSS AIC F value Pr(>F)
## <none> 42.687 -190.17
## x5 1 0.27561 42.412 -189.18 0.9683 0.3267
add1(mod5,~.+x1*x2+x1*x3+x1*x4+x1*x5+x1*x6+x2*x3+x2*x4+x2*x5+x2*x6+x3*x4+x3*x5+x3*x6+x4*x5+x4*x6+x5*x6, test='F')
## Single term additions
##
## Model:
## y ~ x1 + x4 + x2 + x3 + x6
## Df Sum of Sq RSS AIC F value Pr(>F)
## <none> 42.687 -190.17
## x5 1 0.2756 42.412 -189.18 0.9683 0.3267089
## x1:x2 1 5.0857 37.602 -207.96 20.1527 1.422e-05 ***
## x1:x3 1 4.2664 38.421 -204.60 16.5456 7.671e-05 ***
## x1:x4 1 1.6220 41.065 -194.21 5.8852 0.0164632 *
## x1:x6 1 1.2775 41.410 -192.91 4.5966 0.0336580 *
## x2:x3 1 7.2998 35.388 -217.43 30.7360 1.310e-07 ***
## x4:x2 1 2.5442 40.143 -197.75 9.4433 0.0025206 **
## x2:x6 1 3.7156 38.972 -202.38 14.2056 0.0002356 ***
## x4:x3 1 1.6189 41.068 -194.20 5.8737 0.0165675 *
## x3:x6 1 0.8436 41.844 -191.28 3.0039 0.0851320 .
## x4:x6 1 0.3380 42.349 -189.41 1.1891 0.2772766
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
mod6 = update(mod5,~.+x2*x3)
summary(mod6)
##
## Call:
## lm(formula = y ~ x1 + x4 + x2 + x3 + x6 + x2:x3)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.63890 -0.23831 0.02109 0.33968 1.26013
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.9233 0.4121 9.520 < 2e-16 ***
## x1 0.6484 0.1982 3.272 0.001326 **
## x4 1.6121 0.3352 4.810 3.67e-06 ***
## x2 -0.6932 0.3918 -1.769 0.078927 .
## x3 -2.6326 0.7367 -3.573 0.000475 ***
## x6 0.2696 0.5006 0.538 0.591068
## x2:x3 3.2111 0.5792 5.544 1.31e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.4873 on 149 degrees of freedom
## Multiple R-squared: 0.8157, Adjusted R-squared: 0.8083
## F-statistic: 109.9 on 6 and 149 DF, p-value: < 2.2e-16
mod7 = update(mod6,~.-x6)
summary(mod7)
##
## Call:
## lm(formula = y ~ x1 + x4 + x2 + x3 + x2:x3)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.6411 -0.2478 0.0184 0.3616 1.2651
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.9894 0.3925 10.164 < 2e-16 ***
## x1 0.6598 0.1966 3.357 0.001000 **
## x4 1.6826 0.3078 5.466 1.87e-07 ***
## x2 -0.7691 0.3647 -2.109 0.036624 *
## x3 -2.7303 0.7123 -3.833 0.000186 ***
## x2:x3 3.3064 0.5502 6.009 1.35e-08 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.4862 on 150 degrees of freedom
## Multiple R-squared: 0.8154, Adjusted R-squared: 0.8092
## F-statistic: 132.5 on 5 and 150 DF, p-value: < 2.2e-16
add1(mod7,~.+x1*x2+x1*x3+x1*x4+x1*x5+x1*x6+x2*x4+x2*x5+x2*x6+x3*x4+x3*x5+x3*x6+x4*x5+x4*x6+x5*x6, test='F')
## Single term additions
##
## Model:
## y ~ x1 + x4 + x2 + x3 + x2:x3
## Df Sum of Sq RSS AIC F value Pr(>F)
## <none> 35.456 -219.12
## x5 1 0.034508 35.422 -217.27 0.1452 0.7038
## x6 1 0.068859 35.388 -217.43 0.2899 0.5911
## x1:x2 1 0.030737 35.426 -217.26 0.1293 0.7197
## x1:x3 1 0.087048 35.369 -217.51 0.3667 0.5457
## x1:x4 1 0.083232 35.373 -217.49 0.3506 0.5547
## x4:x2 1 0.105871 35.351 -217.59 0.4462 0.5052
## x4:x3 1 0.004253 35.452 -217.14 0.0179 0.8938
#summary(mod8)
# b)AIC:
mod.i = lm(y~(x1+x2+x3+x4+x5+x6)^2)
step(mod0,scope = list(lower=mod0,upper=mod.i))
## Start: AIC=34.43
## y ~ 1
##
## Df Sum of Sq RSS AIC
## + x1 1 121.040 71.011 -118.776
## + x3 1 116.809 75.242 -109.747
## + x2 1 115.964 76.087 -108.005
## + x4 1 61.686 130.365 -24.005
## + x6 1 28.557 163.493 11.319
## <none> 192.051 34.433
## + x5 1 1.104 190.946 35.533
##
## Step: AIC=-118.78
## y ~ x1
##
## Df Sum of Sq RSS AIC
## + x4 1 15.845 55.166 -156.164
## + x2 1 14.108 56.903 -151.327
## + x3 1 8.649 62.361 -137.038
## + x6 1 4.639 66.372 -127.314
## + x5 1 3.738 67.273 -125.212
## <none> 71.011 -118.776
## - x1 1 121.040 192.051 34.433
##
## Step: AIC=-156.16
## y ~ x1 + x4
##
## Df Sum of Sq RSS AIC
## + x2 1 7.837 47.329 -178.067
## + x3 1 5.709 49.457 -171.207
## + x1:x4 1 0.901 54.265 -156.733
## <none> 55.166 -156.164
## + x6 1 0.545 54.621 -155.713
## + x5 1 0.457 54.709 -155.461
## - x4 1 15.845 71.011 -118.776
## - x1 1 75.199 130.365 -24.005
##
## Step: AIC=-178.07
## y ~ x1 + x4 + x2
##
## Df Sum of Sq RSS AIC
## + x1:x2 1 6.5207 40.808 -199.19
## + x2:x4 1 3.4675 43.861 -187.94
## + x3 1 3.3363 43.993 -187.47
## + x1:x4 1 2.0414 45.287 -182.94
## + x6 1 1.6519 45.677 -181.61
## + x5 1 0.7866 46.542 -178.68
## <none> 47.329 -178.07
## - x2 1 7.8369 55.166 -156.16
## - x4 1 9.5743 56.903 -151.33
## - x1 1 17.2282 64.557 -131.64
##
## Step: AIC=-199.19
## y ~ x1 + x4 + x2 + x1:x2
##
## Df Sum of Sq RSS AIC
## + x3 1 3.1524 37.656 -209.73
## + x2:x4 1 0.7128 40.095 -199.94
## <none> 40.808 -199.19
## + x5 1 0.1475 40.661 -197.76
## + x6 1 0.1377 40.670 -197.72
## + x1:x4 1 0.0731 40.735 -197.47
## - x1:x2 1 6.5207 47.329 -178.07
## - x4 1 7.4742 48.282 -174.96
##
## Step: AIC=-209.73
## y ~ x1 + x4 + x2 + x3 + x1:x2
##
## Df Sum of Sq RSS AIC
## + x2:x3 1 2.2301 35.426 -217.26
## + x2:x4 1 0.7140 36.942 -210.72
## + x1:x3 1 0.5242 37.132 -209.92
## <none> 37.656 -209.73
## + x3:x4 1 0.2800 37.376 -208.90
## + x1:x4 1 0.2102 37.446 -208.61
## + x5 1 0.1029 37.553 -208.16
## + x6 1 0.0542 37.602 -207.96
## - x3 1 3.1524 40.808 -199.19
## - x1:x2 1 6.3368 43.993 -187.47
## - x4 1 6.6508 44.307 -186.36
##
## Step: AIC=-217.26
## y ~ x1 + x4 + x2 + x3 + x1:x2 + x2:x3
##
## Df Sum of Sq RSS AIC
## - x1:x2 1 0.0307 35.456 -219.12
## <none> 35.426 -217.26
## + x1:x3 1 0.1298 35.296 -215.83
## + x1:x4 1 0.1027 35.323 -215.71
## + x6 1 0.0953 35.330 -215.68
## + x2:x4 1 0.0936 35.332 -215.67
## + x5 1 0.0354 35.390 -215.41
## + x3:x4 1 0.0031 35.423 -215.27
## - x2:x3 1 2.2301 37.656 -209.73
## - x4 1 7.0798 42.505 -190.84
##
## Step: AIC=-219.12
## y ~ x1 + x4 + x2 + x3 + x2:x3
##
## Df Sum of Sq RSS AIC
## <none> 35.456 -219.12
## + x2:x4 1 0.1059 35.351 -217.59
## + x1:x3 1 0.0870 35.369 -217.51
## + x1:x4 1 0.0832 35.373 -217.49
## + x6 1 0.0689 35.388 -217.43
## + x5 1 0.0345 35.422 -217.27
## + x1:x2 1 0.0307 35.426 -217.26
## + x3:x4 1 0.0043 35.452 -217.14
## - x1 1 2.6630 38.119 -209.82
## - x4 1 7.0631 42.519 -192.78
## - x2:x3 1 8.5362 43.993 -187.47
##
## Call:
## lm(formula = y ~ x1 + x4 + x2 + x3 + x2:x3)
##
## Coefficients:
## (Intercept) x1 x4 x2 x3 x2:x3
## 3.9894 0.6598 1.6826 -0.7691 -2.7303 3.3064
# c)BIC:
step(mod0,scope = list(lower=mod0,upper=mod.i), k=log(n))
## Start: AIC=37.48
## y ~ 1
##
## Df Sum of Sq RSS AIC
## + x1 1 121.040 71.011 -112.676
## + x3 1 116.809 75.242 -103.647
## + x2 1 115.964 76.087 -101.905
## + x4 1 61.686 130.365 -17.906
## + x6 1 28.557 163.493 17.418
## <none> 192.051 37.483
## + x5 1 1.104 190.946 41.633
##
## Step: AIC=-112.68
## y ~ x1
##
## Df Sum of Sq RSS AIC
## + x4 1 15.845 55.166 -147.015
## + x2 1 14.108 56.903 -142.177
## + x3 1 8.649 62.361 -127.888
## + x6 1 4.639 66.372 -118.165
## + x5 1 3.738 67.273 -116.062
## <none> 71.011 -112.676
## - x1 1 121.040 192.051 37.483
##
## Step: AIC=-147.01
## y ~ x1 + x4
##
## Df Sum of Sq RSS AIC
## + x2 1 7.837 47.329 -165.867
## + x3 1 5.709 49.457 -159.007
## <none> 55.166 -147.015
## + x1:x4 1 0.901 54.265 -144.534
## + x6 1 0.545 54.621 -143.514
## + x5 1 0.457 54.709 -143.261
## - x4 1 15.845 71.011 -112.676
## - x1 1 75.199 130.365 -17.906
##
## Step: AIC=-165.87
## y ~ x1 + x4 + x2
##
## Df Sum of Sq RSS AIC
## + x1:x2 1 6.5207 40.808 -183.94
## + x2:x4 1 3.4675 43.861 -172.69
## + x3 1 3.3363 43.993 -172.22
## + x1:x4 1 2.0414 45.287 -167.70
## + x6 1 1.6519 45.677 -166.36
## <none> 47.329 -165.87
## + x5 1 0.7866 46.542 -163.43
## - x2 1 7.8369 55.166 -147.01
## - x4 1 9.5743 56.903 -142.18
## - x1 1 17.2282 64.557 -122.49
##
## Step: AIC=-183.94
## y ~ x1 + x4 + x2 + x1:x2
##
## Df Sum of Sq RSS AIC
## + x3 1 3.1524 37.656 -191.44
## <none> 40.808 -183.94
## + x2:x4 1 0.7128 40.095 -181.64
## + x5 1 0.1475 40.661 -179.46
## + x6 1 0.1377 40.670 -179.42
## + x1:x4 1 0.0731 40.735 -179.17
## - x1:x2 1 6.5207 47.329 -165.87
## - x4 1 7.4742 48.282 -162.76
##
## Step: AIC=-191.43
## y ~ x1 + x4 + x2 + x3 + x1:x2
##
## Df Sum of Sq RSS AIC
## + x2:x3 1 2.2301 35.426 -195.91
## <none> 37.656 -191.44
## + x2:x4 1 0.7140 36.942 -189.37
## + x1:x3 1 0.5242 37.132 -188.57
## + x3:x4 1 0.2800 37.376 -187.55
## + x1:x4 1 0.2102 37.446 -187.26
## + x5 1 0.1029 37.553 -186.81
## + x6 1 0.0542 37.602 -186.61
## - x3 1 3.1524 40.808 -183.94
## - x1:x2 1 6.3368 43.993 -172.22
## - x4 1 6.6508 44.307 -171.11
##
## Step: AIC=-195.91
## y ~ x1 + x4 + x2 + x3 + x1:x2 + x2:x3
##
## Df Sum of Sq RSS AIC
## - x1:x2 1 0.0307 35.456 -200.82
## <none> 35.426 -195.91
## - x2:x3 1 2.2301 37.656 -191.44
## + x1:x3 1 0.1298 35.296 -191.43
## + x1:x4 1 0.1027 35.323 -191.31
## + x6 1 0.0953 35.330 -191.28
## + x2:x4 1 0.0936 35.332 -191.27
## + x5 1 0.0354 35.390 -191.01
## + x3:x4 1 0.0031 35.423 -190.87
## - x4 1 7.0798 42.505 -172.54
##
## Step: AIC=-200.82
## y ~ x1 + x4 + x2 + x3 + x2:x3
##
## Df Sum of Sq RSS AIC
## <none> 35.456 -200.82
## + x2:x4 1 0.1059 35.351 -196.24
## + x1:x3 1 0.0870 35.369 -196.16
## + x1:x4 1 0.0832 35.373 -196.14
## + x6 1 0.0689 35.388 -196.08
## + x5 1 0.0345 35.422 -195.93
## + x1:x2 1 0.0307 35.426 -195.91
## + x3:x4 1 0.0043 35.452 -195.79
## - x1 1 2.6630 38.119 -194.57
## - x4 1 7.0631 42.519 -177.53
## - x2:x3 1 8.5362 43.993 -172.22
##
## Call:
## lm(formula = y ~ x1 + x4 + x2 + x3 + x2:x3)
##
## Coefficients:
## (Intercept) x1 x4 x2 x3 x2:x3
## 3.9894 0.6598 1.6826 -0.7691 -2.7303 3.3064
# NOT RUN {
AIC(mod0)
## [1] 479.1416
AIC(mod1)
## [1] 325.9328
AIC(mod2)
## [1] 288.5446
AIC(mod3)
## [1] 266.6419
AIC(mod4)
## [1] 257.2385
AIC(mod5)
## [1] 254.54
AIC(mod6)
## [1] 227.2833
AIC(mod7)
## [1] 225.5866
#stopifnot(all.equal(AIC(mod0), AIC(logLik(mod0))))
BIC(mod7)
## [1] 246.9356
#lm2 <- update(lm1, . ~ . -Examination)
#AIC(lm1, lm2)
#BIC(lm1, lm2)
# }
# Model after Step-wise regression:
mod8 = lm(y~x1+x4+x2+x3+x2*x3)
mod = regsubsets(y~(x1+x2+x3+x4+x5+x6)^2,data=dat,nvmax=8)
summary(mod)
## Subset selection object
## Call: regsubsets.formula(y ~ (x1 + x2 + x3 + x4 + x5 + x6)^2, data = dat,
## nvmax = 8)
## 21 Variables (and intercept)
## Forced in Forced out
## x1 FALSE FALSE
## x2 FALSE FALSE
## x3 FALSE FALSE
## x4 FALSE FALSE
## x5 FALSE FALSE
## x6 FALSE FALSE
## x1:x2 FALSE FALSE
## x1:x3 FALSE FALSE
## x1:x4 FALSE FALSE
## x1:x5 FALSE FALSE
## x1:x6 FALSE FALSE
## x2:x3 FALSE FALSE
## x2:x4 FALSE FALSE
## x2:x5 FALSE FALSE
## x2:x6 FALSE FALSE
## x3:x4 FALSE FALSE
## x3:x5 FALSE FALSE
## x3:x6 FALSE FALSE
## x4:x5 FALSE FALSE
## x4:x6 FALSE FALSE
## x5:x6 FALSE FALSE
## 1 subsets of each size up to 8
## Selection Algorithm: exhaustive
## x1 x2 x3 x4 x5 x6 x1:x2 x1:x3 x1:x4 x1:x5 x1:x6 x2:x3 x2:x4
## 1 ( 1 ) " " " " " " " " " " " " " " " " " " " " " " "*" " "
## 2 ( 1 ) " " " " " " " " " " " " " " " " "*" " " " " "*" " "
## 3 ( 1 ) " " " " "*" " " " " " " " " " " "*" " " " " "*" " "
## 4 ( 1 ) " " " " " " " " " " " " " " " " "*" " " " " "*" " "
## 5 ( 1 ) " " "*" " " " " " " " " " " " " "*" " " " " "*" " "
## 6 ( 1 ) " " " " " " "*" " " " " " " " " " " "*" " " "*" " "
## 7 ( 1 ) " " "*" " " "*" " " " " " " " " " " "*" " " "*" " "
## 8 ( 1 ) " " "*" "*" "*" " " " " " " " " " " "*" " " "*" " "
## x2:x5 x2:x6 x3:x4 x3:x5 x3:x6 x4:x5 x4:x6 x5:x6
## 1 ( 1 ) " " " " " " " " " " " " " " " "
## 2 ( 1 ) " " " " " " " " " " " " " " " "
## 3 ( 1 ) " " " " " " " " " " " " " " " "
## 4 ( 1 ) " " "*" " " " " "*" " " " " " "
## 5 ( 1 ) " " "*" " " " " "*" " " " " " "
## 6 ( 1 ) " " "*" " " " " "*" "*" " " " "
## 7 ( 1 ) " " "*" " " " " "*" "*" " " " "
## 8 ( 1 ) " " "*" " " " " "*" "*" " " " "
sm = summary(mod)
sm$which
## (Intercept) x1 x2 x3 x4 x5 x6 x1:x2 x1:x3 x1:x4 x1:x5 x1:x6
## 1 TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## 2 TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE FALSE FALSE
## 3 TRUE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE TRUE FALSE FALSE
## 4 TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE FALSE FALSE
## 5 TRUE FALSE TRUE FALSE FALSE FALSE FALSE FALSE FALSE TRUE FALSE FALSE
## 6 TRUE FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE TRUE FALSE
## 7 TRUE FALSE TRUE FALSE TRUE FALSE FALSE FALSE FALSE FALSE TRUE FALSE
## 8 TRUE FALSE TRUE TRUE TRUE FALSE FALSE FALSE FALSE FALSE TRUE FALSE
## x2:x3 x2:x4 x2:x5 x2:x6 x3:x4 x3:x5 x3:x6 x4:x5 x4:x6 x5:x6
## 1 TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## 2 TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## 3 TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## 4 TRUE FALSE FALSE TRUE FALSE FALSE TRUE FALSE FALSE FALSE
## 5 TRUE FALSE FALSE TRUE FALSE FALSE TRUE FALSE FALSE FALSE
## 6 TRUE FALSE FALSE TRUE FALSE FALSE TRUE TRUE FALSE FALSE
## 7 TRUE FALSE FALSE TRUE FALSE FALSE TRUE TRUE FALSE FALSE
## 8 TRUE FALSE FALSE TRUE FALSE FALSE TRUE TRUE FALSE FALSE
sm$adjr2
## [1] 0.7448183 0.7970237 0.8067916 0.8124593 0.8147822 0.8161633 0.8202197
## [8] 0.8217300
sm$adjr2[6]
## [1] 0.8161633
sm$adjr2[8]
## [1] 0.82173
rss = sm$rss
mses = c(rss[1]/(n-2), rss[2]/(n-3), rss[3]/(n-4), rss[4]/(n-5), rss[5]/(n-6), rss[6]/(n-7), rss[7]/(n-8), rss[8]/(n-9))
mses
## [1] 0.3161793 0.2514948 0.2393921 0.2323697 0.2294915 0.2277803 0.2227543
## [8] 0.2208829
sm$cp
## [1] 61.943043 19.069203 11.881227 8.170178 7.252268 7.123605 4.854404
## [8] 4.666938
sm$cp[6]
## [1] 7.123605
mod.sub = lm(y~x1+x4+x2+x3+x5+x6+x1*x5+x2*x3+x2*x6+x3*x6+x4*x5)
plot(mod.sub)
hist(resid(mod.sub))
# Model Assumptions
summary(mod8)
##
## Call:
## lm(formula = y ~ x1 + x4 + x2 + x3 + x2 * x3)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.6411 -0.2478 0.0184 0.3616 1.2651
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.9894 0.3925 10.164 < 2e-16 ***
## x1 0.6598 0.1966 3.357 0.001000 **
## x4 1.6826 0.3078 5.466 1.87e-07 ***
## x2 -0.7691 0.3647 -2.109 0.036624 *
## x3 -2.7303 0.7123 -3.833 0.000186 ***
## x2:x3 3.3064 0.5502 6.009 1.35e-08 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.4862 on 150 degrees of freedom
## Multiple R-squared: 0.8154, Adjusted R-squared: 0.8092
## F-statistic: 132.5 on 5 and 150 DF, p-value: < 2.2e-16
plot(mod8)
#Model Assumptions
summary(mod8)
##
## Call:
## lm(formula = y ~ x1 + x4 + x2 + x3 + x2 * x3)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.6411 -0.2478 0.0184 0.3616 1.2651
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.9894 0.3925 10.164 < 2e-16 ***
## x1 0.6598 0.1966 3.357 0.001000 **
## x4 1.6826 0.3078 5.466 1.87e-07 ***
## x2 -0.7691 0.3647 -2.109 0.036624 *
## x3 -2.7303 0.7123 -3.833 0.000186 ***
## x2:x3 3.3064 0.5502 6.009 1.35e-08 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.4862 on 150 degrees of freedom
## Multiple R-squared: 0.8154, Adjusted R-squared: 0.8092
## F-statistic: 132.5 on 5 and 150 DF, p-value: < 2.2e-16
hist(resid(mod8))
shapiro.test(resid(mod8))
##
## Shapiro-Wilk normality test
##
## data: resid(mod8)
## W = 0.9832, p-value = 0.05482
dat2 = dat[-c(155),]
y = dat2$Score
x1 = dat2$GDP.per.capita
x2 = dat2$Social.support
x3 = dat2$Healthy.life.expectancy
x4 = dat2$Freedom.to.make.life.choices
x5 = dat2$Generosity
x6 = dat2$Perceptions.of.corruption
mod9 = lm(y~x1+x4+x2+x3+x2*x3)
summary(mod9)
##
## Call:
## lm(formula = y ~ x1 + x4 + x2 + x3 + x2 * x3)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.60976 -0.22497 0.02944 0.30369 1.24555
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 4.9521 0.5093 9.724 < 2e-16 ***
## x1 0.6810 0.1921 3.544 0.000526 ***
## x4 1.6863 0.3006 5.609 9.61e-08 ***
## x2 -1.6227 0.4639 -3.498 0.000619 ***
## x3 -4.1455 0.8526 -4.862 2.92e-06 ***
## x2:x3 4.4766 0.6744 6.638 5.56e-10 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.4749 on 149 degrees of freedom
## Multiple R-squared: 0.82, Adjusted R-squared: 0.8139
## F-statistic: 135.7 on 5 and 149 DF, p-value: < 2.2e-16
plot(mod9)
hist(resid(mod9))
shapiro.test(resid(mod9))
##
## Shapiro-Wilk normality test
##
## data: resid(mod9)
## W = 0.9791, p-value = 0.01868
boxc = boxcox(y~x1+x4+x2+x3+x2*x3, data = dat2, lambda = seq(-2, 2, 0.1))
lambda = boxc$x[which.max(boxc$y)]
lambda
## [1] 1.434343
mod10 = lm(y^lambda~x1+x4+x2+x3+x2*x3)
shapiro.test(resid(mod10))
##
## Shapiro-Wilk normality test
##
## data: resid(mod10)
## W = 0.98361, p-value = 0.06295
hist(resid(mod10))
plot(mod10)
# Begin full vs reduced comparison
modred <- lm(y~x1+x2+x3+x4+x2*x3)
modfull <- lm(y~(x1+x2+x3+x4+x5+x6)^2)
anova(modred,modfull)
## Analysis of Variance Table
##
## Model 1: y ~ x1 + x2 + x3 + x4 + x2 * x3
## Model 2: y ~ (x1 + x2 + x3 + x4 + x5 + x6)^2
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 149 33.597
## 2 133 28.564 16 5.033 1.4646 0.1222
modred2 <- mod10
anova(modred2,modfull)
## Warning in anova.lmlist(object, ...): models with response '"y"' removed because
## response differs from model 1
## Analysis of Variance Table
##
## Response: y^lambda
## Df Sum Sq Mean Sq F value Pr(>F)
## x1 1 1024.23 1024.23 527.991 < 2.2e-16 ***
## x4 1 142.49 142.49 73.454 1.209e-14 ***
## x2 1 66.24 66.24 34.148 3.099e-08 ***
## x3 1 30.79 30.79 15.873 0.0001057 ***
## x2:x3 1 110.17 110.17 56.793 4.353e-12 ***
## Residuals 149 289.04 1.94
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
# Model after Step-wise regression:
mod8 = lm(y~x1+x4+x2+x3+x2*x3)
# Best Subsets Regression
mod = regsubsets(cbind(x1,x2,x3,x4,x5,x6),y)
sm = summary(mod)
sm$which
## (Intercept) x1 x2 x3 x4 x5 x6
## 1 TRUE TRUE FALSE FALSE FALSE FALSE FALSE
## 2 TRUE TRUE FALSE FALSE TRUE FALSE FALSE
## 3 TRUE TRUE TRUE FALSE TRUE FALSE FALSE
## 4 TRUE TRUE TRUE TRUE TRUE FALSE FALSE
## 5 TRUE TRUE TRUE TRUE TRUE FALSE TRUE
## 6 TRUE TRUE TRUE TRUE TRUE TRUE TRUE
sm$adjr2
## [1] 0.6177759 0.7008756 0.7433231 0.7605097 0.7666651 0.7664649
rss = sm$rss
mses = c(rss[1]/(n-2), rss[2]/(n-3), rss[3]/(n-4), rss[4]/(n-5), rss[5]/(n-6), rss[6]/(n-7))
mses
## [1] 0.4601643 0.3601042 0.3089899 0.2882878 0.2808658 0.2810941
sm$cp
## [1] 99.413265 45.689832 18.963116 8.825016 5.872250 7.000000
# Model after Best Subsets Regression
mod.sub = lm(y~x1+x4+x2+x3+x6)
# Model Assumptions
summary(mod8)
##
## Call:
## lm(formula = y ~ x1 + x4 + x2 + x3 + x2 * x3)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.60976 -0.22497 0.02944 0.30369 1.24555
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 4.9521 0.5093 9.724 < 2e-16 ***
## x1 0.6810 0.1921 3.544 0.000526 ***
## x4 1.6863 0.3006 5.609 9.61e-08 ***
## x2 -1.6227 0.4639 -3.498 0.000619 ***
## x3 -4.1455 0.8526 -4.862 2.92e-06 ***
## x2:x3 4.4766 0.6744 6.638 5.56e-10 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.4749 on 149 degrees of freedom
## Multiple R-squared: 0.82, Adjusted R-squared: 0.8139
## F-statistic: 135.7 on 5 and 149 DF, p-value: < 2.2e-16
plot(mod8)
hist(resid(mod8))
shapiro.test(resid(mod8))
##
## Shapiro-Wilk normality test
##
## data: resid(mod8)
## W = 0.9791, p-value = 0.01868
dat2 = dat[-c(155),]
y = dat2$Score
x1 = dat2$GDP.per.capita
x2 = dat2$Social.support
x3 = dat2$Healthy.life.expectancy
x4 = dat2$Freedom.to.make.life.choices
x5 = dat2$Generosity
x6 = dat2$Perceptions.of.corruption
mod9 = lm(y~x1+x4+x2+x3+x2*x3)
summary(mod9)
##
## Call:
## lm(formula = y ~ x1 + x4 + x2 + x3 + x2 * x3)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.60976 -0.22497 0.02944 0.30369 1.24555
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 4.9521 0.5093 9.724 < 2e-16 ***
## x1 0.6810 0.1921 3.544 0.000526 ***
## x4 1.6863 0.3006 5.609 9.61e-08 ***
## x2 -1.6227 0.4639 -3.498 0.000619 ***
## x3 -4.1455 0.8526 -4.862 2.92e-06 ***
## x2:x3 4.4766 0.6744 6.638 5.56e-10 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.4749 on 149 degrees of freedom
## Multiple R-squared: 0.82, Adjusted R-squared: 0.8139
## F-statistic: 135.7 on 5 and 149 DF, p-value: < 2.2e-16
plot(mod9)
shapiro.test(resid(mod9))
##
## Shapiro-Wilk normality test
##
## data: resid(mod9)
## W = 0.9791, p-value = 0.01868
boxc = boxcox(y~x1+x4+x2+x3+x2*x3, data = dat2, lambda = seq(-2, 2, 0.1))
lambda = boxc$x[which.max(boxc$y)]
mod10 = lm(y^lambda~x1+x4+x2+x3+x2*x3)
shapiro.test(resid(mod10))
##
## Shapiro-Wilk normality test
##
## data: resid(mod10)
## W = 0.98361, p-value = 0.06295
plot(mod10)
# Begin full vs reduced comparison
modred <- lm(y~x1+x2+x3+x4+x2*x3)
modfull <- lm(y~(x1+x2+x3+x4+x5+x6+I(x6^2))^2)
anova(modred,modfull)
## Analysis of Variance Table
##
## Model 1: y ~ x1 + x2 + x3 + x4 + x2 * x3
## Model 2: y ~ (x1 + x2 + x3 + x4 + x5 + x6 + I(x6^2))^2
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 149 33.597
## 2 126 26.567 23 7.0306 1.4498 0.1009
# Conclude reduced is better
summary(mod9)
##
## Call:
## lm(formula = y ~ x1 + x4 + x2 + x3 + x2 * x3)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.60976 -0.22497 0.02944 0.30369 1.24555
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 4.9521 0.5093 9.724 < 2e-16 ***
## x1 0.6810 0.1921 3.544 0.000526 ***
## x4 1.6863 0.3006 5.609 9.61e-08 ***
## x2 -1.6227 0.4639 -3.498 0.000619 ***
## x3 -4.1455 0.8526 -4.862 2.92e-06 ***
## x2:x3 4.4766 0.6744 6.638 5.56e-10 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.4749 on 149 degrees of freedom
## Multiple R-squared: 0.82, Adjusted R-squared: 0.8139
## F-statistic: 135.7 on 5 and 149 DF, p-value: < 2.2e-16
cooks.distance(mod7)
## 1 2 3 4 5 6
## 1.049689e-02 5.170036e-03 1.432856e-03 6.995258e-04 5.120677e-03 2.407868e-03
## 7 8 9 10 11 12
## 3.221363e-03 9.796633e-04 1.200415e-03 2.777868e-03 3.148469e-04 9.753133e-03
## 13 14 15 16 17 18
## 8.193624e-03 2.948024e-04 1.325853e-03 2.568323e-05 1.428595e-03 5.407540e-04
## 19 20 21 22 23 24
## 4.087471e-03 1.631698e-03 1.265441e-02 7.375629e-04 3.874418e-03 2.029278e-04
## 25 26 27 28 29 30
## 8.252020e-04 2.278081e-03 1.151198e-02 3.195050e-03 1.799134e-04 1.789754e-03
## 31 32 33 34 35 36
## 1.586001e-05 2.074131e-03 5.996229e-05 3.058460e-02 8.424565e-03 7.534879e-04
## 37 38 39 40 41 42
## 1.743230e-04 4.888054e-07 5.936122e-04 8.233536e-05 9.565630e-05 4.943048e-04
## 43 44 45 46 47 48
## 2.110463e-04 7.739605e-03 7.194322e-03 3.956601e-03 5.453470e-05 1.757552e-03
## 49 50 51 52 53 54
## 2.986060e-04 5.479707e-04 3.072500e-04 1.763946e-04 3.473707e-04 6.320577e-03
## 55 56 57 58 59 60
## 4.475588e-03 3.395319e-04 1.268331e-04 1.188379e-02 4.935722e-03 2.818429e-04
## 61 62 63 64 65 66
## 2.652332e-03 4.013287e-04 7.617066e-04 2.008236e-03 4.582002e-06 9.894691e-03
## 67 68 69 70 71 72
## 2.094039e-02 8.913423e-05 3.905539e-04 1.573656e-05 5.413600e-03 9.058306e-05
## 73 74 75 76 77 78
## 4.592831e-06 1.177645e-02 4.326127e-04 2.589701e-02 1.958027e-03 1.467337e-03
## 79 80 81 82 83 84
## 2.715219e-04 2.997218e-03 2.103827e-03 2.719602e-03 1.499058e-03 4.984239e-04
## 85 86 87 88 89 90
## 2.892667e-02 3.557102e-03 5.185894e-03 7.380709e-03 2.932327e-02 3.366936e-07
## 91 92 93 94 95 96
## 6.583537e-05 2.402727e-04 3.540444e-03 1.042865e-02 4.942719e-04 9.917056e-03
## 97 98 99 100 101 102
## 1.467298e-02 4.435309e-03 1.804694e-02 1.111516e-04 1.801153e-03 4.371052e-02
## 103 104 105 106 107 108
## 3.276406e-03 5.728701e-04 9.504668e-04 4.376866e-03 1.603222e-04 1.577864e-02
## 109 110 111 112 113 114
## 6.498007e-03 1.390536e-04 6.962798e-04 1.101898e-02 4.022505e-03 1.156969e-02
## 115 116 117 118 119 120
## 5.078605e-03 1.598727e-03 3.398840e-04 2.187656e-03 6.018901e-03 1.005339e-03
## 121 122 123 124 125 126
## 1.188549e-04 5.552154e-03 1.279468e-04 5.047628e-04 4.065722e-03 1.148529e-03
## 127 128 129 130 131 132
## 8.611686e-03 7.412165e-04 3.146845e-03 1.431296e-02 9.767987e-03 1.111701e-02
## 133 134 135 136 137 138
## 1.978575e-02 1.068321e-04 3.930002e-04 4.551653e-04 4.128009e-03 4.276355e-03
## 139 140 141 142 143 144
## 1.309776e-03 1.450567e-02 4.864694e-04 2.035824e-03 9.571560e-05 1.055017e-02
## 145 146 147 148 149 150
## 2.412243e-03 1.105206e-02 3.078043e-04 8.366687e-02 2.301207e-05 1.675004e-02
## 151 152 153 154 155 156
## 1.981495e-02 6.514437e-02 2.059902e-02 4.258839e-03 1.052905e+00 3.145728e-02
rstudent(mod7)
## 1 2 3 4 5 6
## 1.32542081 0.94336423 0.44926717 0.29550739 1.06599253 0.64343611
## 7 8 9 10 11 12
## 0.85428073 0.39776172 0.47801344 0.84605808 0.23508987 1.47701345
## 13 14 15 16 17 18
## 1.38259631 0.22053300 0.55142529 -0.06947194 0.67328962 0.39099678
## 19 20 21 22 23 24
## 1.01908911 0.80439196 1.11208292 -0.40367442 1.60561686 -0.21867099
## 25 26 27 28 29 30
## 0.49155886 1.01388509 1.88791968 0.76907638 -0.11447083 -0.54912230
## 31 32 33 34 35 36
## -0.08220691 0.97811450 -0.15380530 -2.07942901 1.84985134 -0.29331932
## 37 38 39 40 41 42
## -0.20820005 0.01155022 0.35010075 -0.20481216 0.10403444 0.32069240
## 43 44 45 46 47 48
## 0.31927473 -1.41138591 1.18465802 1.31882261 -0.16697198 0.77895275
## 49 50 51 52 53 54
## 0.23370099 0.43961593 -0.19095502 -0.25719221 0.27785986 0.78352594
## 55 56 57 58 59 60
## -1.24618086 -0.26521520 -0.24842652 -1.61498213 0.88249219 -0.26726924
## 61 62 63 64 65 66
## 0.93981716 0.26342310 -0.43362764 -0.64943582 0.04699062 -1.78873130
## 67 68 69 70 71 72
## 2.67930272 -0.14619395 0.31071952 -0.06855319 0.98187648 0.17985252
## 73 74 75 76 77 78
## 0.02918037 1.46068161 -0.36088505 -1.99185012 -0.98168783 0.56053802
## 79 80 81 82 83 84
## -0.22156122 -0.74150386 -0.63918041 0.43579213 -0.50329621 -0.50892306
## 85 86 87 88 89 90
## 1.54767141 -0.64063912 -0.96935141 0.92838063 1.50437350 0.01078424
## 91 92 93 94 95 96
## 0.12694374 -0.28099046 -0.77791856 -1.35413392 -0.38483889 1.38071524
## 97 98 99 100 101 102
## -1.91214083 1.16881557 1.32179528 -0.13021248 -0.85651344 1.92008584
## 103 104 105 106 107 108
## 0.85808121 -0.31043968 -0.41235851 -0.69720503 0.10330213 -1.43097377
## 109 110 111 112 113 114
## -0.95293618 -0.17120309 0.40214873 0.83435454 -0.76233404 1.40760747
## 115 116 117 118 119 120
## 0.98794881 -0.56994523 -0.15961107 0.76597840 0.58725806 0.50354755
## 121 122 123 124 125 126
## -0.18525226 0.75890678 0.13097828 -0.24677967 -0.67754214 -0.38813607
## 127 128 129 130 131 132
## 0.90292663 0.33828494 0.67558651 -2.69149175 -1.51632270 0.98638411
## 133 134 135 136 137 138
## -1.70574207 -0.15364272 0.10122431 -0.31705080 -1.01224924 -1.04565708
## 139 140 141 142 143 144
## 0.42951073 -1.27778413 -0.24553825 0.54619093 0.11052160 -0.86797787
## 145 146 147 148 149 150
## 0.43565528 -1.53203749 0.17796384 -3.57911689 -0.03376150 -1.24438600
## 151 152 153 154 155 156
## -1.40507600 -2.26522962 -2.51833924 -0.53757664 -2.87153201 -1.44125372
cooks.distance(modred)
## 1 2 3 4 5 6
## 9.352437e-03 4.125265e-03 5.306230e-04 5.156334e-05 4.499310e-03 1.507651e-03
## 7 8 9 10 11 12
## 2.720219e-03 3.754136e-04 6.721946e-04 2.373646e-03 2.542842e-05 9.779971e-03
## 13 14 15 16 17 18
## 7.799401e-03 7.243498e-05 8.799056e-04 2.987501e-04 1.235026e-03 3.288777e-04
## 19 20 21 22 23 24
## 4.208541e-03 1.555404e-03 1.460258e-02 1.401958e-03 4.560094e-03 5.449076e-04
## 25 26 27 28 29 30
## 8.057834e-04 2.472970e-03 1.437483e-02 3.904167e-03 1.004814e-04 3.153717e-03
## 31 32 33 34 35 36
## 2.203105e-05 2.509647e-03 7.102555e-05 4.228843e-02 1.070879e-02 1.681624e-03
## 37 38 39 40 41 42
## 1.411672e-04 3.949242e-07 1.060533e-03 8.504125e-05 3.450793e-04 5.921786e-04
## 43 44 45 46 47 48
## 2.896399e-04 9.522094e-03 8.859869e-03 5.257599e-03 5.030492e-05 2.383619e-03
## 49 50 51 52 53 54
## 5.605812e-04 7.835154e-04 1.667473e-04 1.272432e-04 4.761753e-04 8.169628e-03
## 55 56 57 58 59 60
## 4.914787e-03 2.638920e-04 8.048048e-05 1.476422e-02 6.766268e-03 1.364433e-04
## 61 62 63 64 65 66
## 3.727472e-03 5.694209e-04 5.716173e-04 1.798515e-03 3.641738e-05 1.133607e-02
## 67 68 69 70 71 72
## 2.153183e-02 1.363149e-05 8.452966e-04 1.945106e-06 7.365396e-03 2.645710e-04
## 73 74 75 76 77 78
## 3.089186e-05 1.522396e-02 3.125018e-04 2.680393e-02 1.859833e-03 2.243804e-03
## 79 80 81 82 83 84
## 1.497213e-04 2.471227e-03 1.846719e-03 4.956494e-03 8.434542e-04 4.035726e-04
## 85 86 87 88 89 90
## 3.010950e-02 2.549067e-03 4.223515e-03 9.945254e-03 3.983148e-02 4.106783e-05
## 91 92 93 94 95 96
## 2.237833e-04 1.142530e-04 2.749101e-03 1.004460e-02 2.394192e-04 8.976609e-03
## 97 98 99 100 101 102
## 1.496382e-02 4.185297e-03 1.291458e-02 2.668525e-07 1.603253e-03 3.826041e-02
## 103 104 105 106 107 108
## 2.773276e-03 3.234901e-04 8.650037e-04 2.648174e-03 1.952933e-03 1.531348e-02
## 109 110 111 112 113 114
## 5.674886e-03 1.446586e-05 1.158194e-03 3.303315e-03 2.808158e-03 1.019760e-02
## 115 116 117 118 119 120
## 5.297769e-03 1.050446e-03 2.375946e-06 1.547831e-03 9.882302e-03 8.023214e-04
## 121 122 123 124 125 126
## 8.082060e-05 7.545695e-03 4.773599e-05 9.065894e-05 3.062802e-03 1.142957e-03
## 127 128 129 130 131 132
## 1.042546e-02 8.634440e-04 1.290339e-03 1.512337e-02 9.404858e-03 4.856524e-03
## 133 134 135 136 137 138
## 1.893781e-02 5.937387e-05 4.057156e-04 3.947745e-04 3.866659e-03 4.449472e-03
## 139 140 141 142 143 144
## 4.125864e-05 1.642097e-02 8.114311e-04 1.382032e-03 1.131568e-04 9.493885e-03
## 145 146 147 148 149 150
## 1.449502e-04 1.090086e-02 2.981156e-05 8.558444e-02 7.707716e-03 2.839985e-02
## 151 152 153 154 155
## 1.828324e-02 6.930700e-02 2.324369e-02 2.466200e-02 9.865568e-02
rstudent(modred)
## 1 2 3 4 5 6
## 1.197388899 0.804773657 0.257762325 0.075134334 0.961812605 0.481962900
## 7 8 9 10 11 12
## 0.762043160 0.234351534 0.343031204 0.758332654 0.063183264 1.474110728
## 13 14 15 16 17 18
## 1.319122538 0.106675322 0.430783342 -0.226399653 0.614634184 0.295363482
## 19 20 21 22 23 24
## 1.033928025 0.778926095 1.191950838 -0.537919547 1.705060609 -0.345945823
## 25 26 27 28 29 30
## 0.485240824 1.055280871 2.044463237 0.845137782 -0.085474252 -0.704753526
## 31 32 33 34 35 36
## -0.096819077 1.059964874 -0.167326066 -2.334530993 2.003604023 -0.429646069
## 37 38 39 40 41 42
## -0.187014756 -0.010367147 0.458325026 -0.208144285 0.195722835 0.350644682
## 43 44 45 46 47 48
## 0.370560696 -1.536162450 1.297868594 1.455530280 -0.160264390 0.884554717
## 49 50 51 52 53 54
## 0.316621216 0.517357136 -0.140115046 -0.216582629 0.324159420 0.885075888
## 55 56 57 58 59 60
## -1.303274044 -0.233044399 -0.194373258 -1.761297395 1.014244317 -0.181903159
## 61 62 63 64 65 66
## 1.072535526 0.312744408 -0.370420305 -0.610514619 0.128409366 -1.894857812
## 67 68 69 70 71 72
## 2.702513839 -0.055941086 0.440588959 -0.023941369 1.120113435 0.294438883
## 73 74 75 76 77 78
## 0.075372932 1.620173635 -0.302428405 -2.026922221 -0.931019372 0.677553637
## 79 80 81 82 83 84
## -0.163250487 -0.660777294 -0.594371531 0.580623316 -0.362427508 -0.443821945
## 85 86 87 88 89 90
## 1.579524106 -0.532015244 -0.835641787 1.062638952 1.716832437 0.114747803
## 91 92 93 94 95 96
## 0.228457366 -0.187364166 -0.664457604 -1.317083565 -0.252311350 1.236199296
## 97 98 99 100 101 102
## -1.926642967 1.107093544 0.973386350 -0.006218343 -0.773810371 1.453673003
## 103 104 105 106 107 108
## 0.766880790 -0.230098623 -0.392710135 -0.518575676 0.346172187 -1.401771537
## 109 110 111 112 113 114
## -0.877757850 -0.053476281 0.507663265 0.399323394 -0.609533351 1.186267361
## 115 116 117 118 119 120
## 1.009171177 -0.443466867 -0.013099783 0.577876993 0.743526908 0.443743047
## 121 122 123 124 125 126
## -0.152106593 0.875964688 0.079660417 -0.101590962 -0.578551234 -0.387127894
## 127 128 129 130 131 132
## 0.989818216 0.364923086 0.377386563 -2.667574662 -1.452494496 0.542968234
## 133 134 135 136 137 138
## -1.639594652 -0.114038492 0.102846629 -0.294636275 -0.967738390 -1.066727642
## 139 140 141 142 143 144
## 0.063639688 -1.356867781 -0.315440627 0.440206999 0.120160186 -0.820130245
## 145 146 147 148 149 150
## -0.086962922 -1.510020662 -0.052252862 -3.597992350 -0.543799886 -1.532878877
## 151 152 153 154 155
## -1.333283752 -2.338747141 -2.649260672 -1.089632336 -2.106785029